Integer quantum Hall effect and Hofstadter's butterfly spectra in three-dimensional metals in external periodic modulations

We propose that Hofstadter's butterfly accompanied by quantum Hall effect that is similar to those predicted to occur in 3D tight-binding systems by Koshino et al. [Phys. Rev. Lett. 86, 1062 (2001)] can be realized in an entirely different system\char22{}3D metals applied with weak external periodic modulations (e.g., acoustic waves). Namely, an effect of two periodic potentials interferes with Landau's quantization due to an applied magnetic field $\mathit{B},$ resulting generally in fractal energy gaps as a function of the tilting angle of $\mathit{B},$ for which the accompanying quantized Hall tensors are computed. The phenomenon arises from the fact that, while the present system has a different physical origin for the butterfly from the 3D tight-binding systems, the mathematical forms are remarkably equivalent.